In the previous problem, using the notation $u _ { i j } = u ( x , \mathrm { t } )$ , and letting $\lambda = c k / h$ , the equation becomes
A) $u _ { i , j + 1 } = \lambda ^ { 2 } { } _ { u i } + 1 , j + \left( 1 + 2 \lambda ^ { 2 } \right) u _ { i j } + \lambda _ { u i - 1 , j - u i j - 1 } ^ { 2 }$
B) $u _ { i , j + 1 } = \lambda ^ { 2 } { } _ { u i } + 1 , j + \left( 1 - 2 \lambda ^ { 2 } \right) u _ { i j } + \lambda _ { u i - 1 , j - u i j - 1 } ^ { 2 }$
C) $u _ { i , j + 1 } = \lambda _ { u i + 1 , j } + ( 1 - \lambda ) u _ { i j } + \lambda _ { u i - 1 , j - u i j - 1 }$
D) $u _ { i , j - 1 } = \lambda ^ { 2 } { } _ { u i } + 1 , j + \left( 1 + 2 \lambda ^ { 2 } \right) u _ { i j } + \lambda _ { u i - 1 , j + u i j - 1 } ^ { 2 }$
E) $u _ { i , j - 1 } = \lambda _ { u i + 1 , j } + ( 1 - 2 \lambda ) u _ { i j } + \lambda _ { u i - 1 , j + u i j - 1 }$

Question

Quizplus · Accepted Answer

The correct equation for a finite difference approximation typically involves a balance of terms from previous time steps and spatial steps. Choice B correctly shows a balance with a central term $u_{ij}$ adjusted by $\lambda^2$, and terms for $u_{i+1,j}$ and $u_{i-1,j}$ that would be part of a spatial difference approximation, with the coefficients correctly reflecting a finite difference scheme for wave or diffusion equations, where $\lambda = ck/h$ is a common form of the Courant number or a similar dimensionless parameter in numerical methods for partial differential equations.

In the Previous Problem, Using the Notation uij=u(x,t)u _ { i j } = u ( x , \mathrm { t } )uij​=u(x,t)

In the Previous Problem, Using the Notation $u _ { i j } = u ( x , \mathrm { t } )$